Most of the case for global warming comes from computer models. The question is, if computer models have trouble predicting the weather a week from now then can they predict the climate fifty years from now?
RealClimate has an answer to this here. This is an influential climate blog. Al Gore recommends it and it has a relationship with Michael Mann, the creator of the Hockey Stick paleo-temperature reconstruction.
To summarize what RealClimate says, weather prediction is hard but climate is easy. Summer and winter are predictable. You can predict that it is cooler at high altitudes. Some local areas have distinct climates such as Northern Europe. Also, the greenhouse effect is demonstrable and carbon dioxide is known to absorb heat.
So - case closed? Not in the least. Yes, the seasons are predictable but this is nothing but statistical averaging. It is not a model that uses all known factors and computes the climate. It is not testable - you cannot roll it forward and backward and compare it to actual measurements (because that's all it is). And you cannot predict change with a statistical average.
It is also possible to predict the same climate using an invalid model. In the Middle Ages philosophers spent a lot of effort computing how the world worked but they started with the assumption that the Sun revolves around the Earth.
For that matter, a 10-year-old could make the same predictions based on nothing but his own experience.
What RealClimate was trying to say and many of the comments say is that local weather is subject to too many random fluctuations to be reliably predictable but these tend to cancel each other out when figured over a large enough area (a hemisphere for example). This allows a simplification in climate models so that it can be reduced to basic forcings. Since CO2 is a forcing, all we have to do is add in the heat gain from additional CO2 and we have the future climate.
Is this true? No. If it was then temperatures for the 20th century would show a smooth climb. Instead there is a climb through the 1930s, a dip that lasted into the 1970s, and a climb again.
The truth is that there are significant unknowns. Not all forcings are known nor is their effect. There are a numerous substances being released into the atmosphere constantly, both from nature and human-derived. These have different effects, some canceling out others.
There are also natural rhythms that are not well-understood. Both the Atlantic and Pacific have multi-decadal oscillations that warm and cool the oceans. El Nino and la Nina cycles are not at all understood but have a major effect. The effects of solar variations may be underestimated.
Then there are what Donald Rumsfeld called "unknown unknowns". If you push a car down a hill and measure it's speed over the first hundred feet you will see it accelerating. From this you would conclude that it would continue to accelerate. In fact, the faster the car goes the more friction is generated. Eventually the friction balances the acceleration and it reaches a stable speed... until you run out of hill.
Unless the driver pops the clutch and the engine starts, speeding the car up even more.
Water vapor and the effect of clouds are the unknown unknowns here. Global warming theory holds that as temperature increases, the amount of water vapor in the atmosphere will increase until it reaches a tipping point. This would be equivalent to popping the clutch. Or it might cause more clouds, trapping even more heat which would also correspond to popping the clutch.
On the other hand, more clouds might reflect heat, stabilizing the temperature, corresponding to friction.
Or there might be other warming factors that have not yet been discovered but are subject to cycles. That would be running out of hill.
So, climate projection is possible if you know all of the forcings but impossible if you don't.
Do we know all of the forcings? No. That's where being able to roll the models forward and backward is important. If the models are accurate then they will match historic climate. They don't. The programmers have never solved this. They have forced their models to match the real world but that means that there is no guarantee that they will continue to match when predicting the future.
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